A Concurrency from the Reflection of the Incircle's Contact Points across the Incenter
Let ABC be a triangle. Let A', B', and C' be the intersections of the incircle with BC, CA, and AB, respectively. Let A'', B'', and C'' be the reflections of A', B', and C' across the incenter. Then AA'', BB'', and CC'' are concurrent.
See problem 18 in Classical Theorems in Plane Geometry.